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Russia Nizhni Novgorod
Section Mechanics
Title On limit cycles, resonance and homoclinic structures in asymmetric pendulum-type equation
Author(-s) Kostromina O.S.a
Affiliations Nizhni Novgorod State Universitya
Abstract Time-periodic perturbations of an asymmetric pendulum-type equation close to an integrable standard equation of a mathematical pendulum are considered. For an autonomous equation, the problem of limit cycles, which reduces to the study of the Poincaré-Pontryagin generating functions, is solved. A partition of the parameter plane into domains with different behavior of the phase curves is constructed. Basic phase portraits for each domain of the obtained partition are given. For a nonautonomous equation, the question of the structure of the resonance zones, to which the solution of the problem of synchronization of oscillations leads, is studied. Averaged equations of the pendulum type, describing the behavior of solutions of the original equation in individual resonance zones, are calculated and analyzed. The global behavior of solutions in cells that do not contain small neighborhoods of unperturbed separatrices is ascertained. Using the analytical Melnikov method and numerical modeling, the basic bifurcations of the nonautonomous equation associated with the appearance of nonrough homoclinic curves are studied. On the plane of the main parameters, a bifurcation diagram for the Poincaré map generated by the original equation, describing different types of homoclinic tangencies of the separatrices of the saddle fixed point, is constructed. Homoclinic zones (those domains of parameters for which homoclinic trajectories to the saddle fixed point exist) with nonsmooth bifurcation boundaries are found.
Keywords pendulum-type equation, limit cycles, resonances, Poincaré homoclinic structures
UDC 517.925.42
MSC 34C15
DOI 10.20537/vm190207
Received 18 March 2019
Language English
Citation Kostromina O.S. On limit cycles, resonance and homoclinic structures in asymmetric pendulum-type equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 228-244.
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